What is (\partial)/(\partial x)((x+2y)/(3x+4y)) ?

The answer to (\partial)/(\partial x)((x+2y)/(3x+4y)) is -(2y)/((3x+4y)^2)

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(\partial)/(\partial x)((x+2y)/(3x+4y))

\frac{\partial}{\partial x} (\frac{x + 2 y}{3 x + 4 y})

- \frac{2 y}{( 3 x + 4 y ) ^{2}}

Solution steps

\frac{\partial}{\partial x} (\frac{x + 2 y}{3 x + 4 y})

Treat

y

as a constant

Apply the Quotient Rule:

(\frac{f}{g})^{'} = \frac{f ^{'} \cdot g - g ^{'} \cdot f}{g ^{2}}

= \frac{\frac{\partial}{\partial x} ( x + 2 y ) ( 3 x + 4 y ) - \frac{\partial}{\partial x} ( 3 x + 4 y ) ( x + 2 y )}{( 3 x + 4 y ) ^{2}}

\frac{\partial}{\partial x} (x + 2 y) = 1

\frac{\partial}{\partial x} (3 x + 4 y) = 3

= \frac{1 \cdot ( 3 x + 4 y ) - 3 ( x + 2 y )}{( 3 x + 4 y ) ^{2}}

Simplify

\frac{1 \cdot ( 3 x + 4 y ) - 3 ( x + 2 y )}{( 3 x + 4 y ) ^{2}} : - \frac{2 y}{( 3 x + 4 y ) ^{2}}

= - \frac{2 y}{( 3 x + 4 y ) ^{2}}

\frac{d}{d θ} (cos (3 θ))

(1 + x + x^{2})^{99}

\int e^{5 x^{2} + 10 x + 10} (x + 1) d x

\frac{d}{d x} (sin (\frac{x}{8}))

x \to 2 π - lim (e^{\frac{1}{x}} - x)

What is (\partial)/(\partial x)((x+2y)/(3x+4y)) ?
The answer to (\partial)/(\partial x)((x+2y)/(3x+4y)) is -(2y)/((3x+4y)^2)